The universal enveloping algebra $U(\mathfrak{ sl}_2)$ of $\mathfrak{ sl}_2$ is a unital associative algebra over $\mathbb C$ generated by $E,F,H$ subject to the relations \begin{ align*} [H,E]=2E, \qquad [H,F]=-2F, \qquad [E,F]=H. \end{ align*} The distinguished central element $$ \Lambda=EF+FE+\frac{ H^2}{ 2} $$ is called the Casimir element of $U(\mathfrak{ sl}_2)$. The universal Hahn algebra $\mathcal H$ is a unital associative algebra over $\mathbb C$ with generators $A,B,C$ and the relations assert that $[A,B]=C$ and each of \begin{ align*} \alpha=[C,A]+2A^2+B, \qquad \beta=[B,C]+4BA+2C \end{ align*} is central in $\mathcal H$. The distinguished central element $$ \Omega=4ABA+B^2-C^2-2\beta A+2(1-\alpha)B $$ is called the Casimir element of $\mathcal H$. By investigating the relationship between the Terwilliger algebras of the hypercube and its halved graph, we discover the algebra homomorphism $\natural:\mathcal H\rightarrow U(\mathfrak{ sl}_2)$ that sends \begin{ eqnarray*} A &\mapsto & \frac{ H}{ 4}, \\ B & \mapsto & \frac{ E^2+F^2+\Lambda-1}{ 4}-\frac{ H^2}{ 8}, \\ C & \mapsto & \frac{ E^2-F^2}{ 4}. \end{ eqnarray*} We determine the image of $\natural$ and show that the kernel of $\natural$ is the two-sided ideal of $\mathcal H$ generated by $\beta$ and $16 \Omega-24 \alpha+3$. By pulling back via $\natural$ each $U(\mathfrak{ sl}_2)$-module can be regarded as an $\mathcal H$-module. For each integer $n\geq 0$ there exists a unique $(n+1)$-dimensional irreducible $U(\mathfrak{ sl}_2)$-module $L_n$ up to isomorphism. We show that the $\mathcal H$-module $L_n$ ($n\geq 1$) is a direct sum of two non-isomorphic irreducible $\mathcal H$-modules.
Categories: math.CO math.RT
